Further Mathematics · Lesson 19

Mechanics: forces and motion

WAECNECO

What the syllabus asks

WAEC: Mechanics: statics (forces, resultant, resolution, equilibrium of a particle, Lami's theorem, moments, parallel forces, friction); dynamics (motion in a straight line, equations of motion, vertical motion under gravity, projectiles, Newton's laws, connected particles, impulse and momentum, work, energy and power)

NECO: Mechanics: statics (forces, resultant, resolution, equilibrium of a particle, Lami's theorem, moments, parallel forces, friction); dynamics (motion in a straight line, equations of motion, vertical motion under gravity, projectiles, Newton's laws, connected particles, impulse and momentum, work, energy and power)

What you will learn

Find resultants by resolving and solve equilibrium problems, including Lami's theorem.
Take moments to find reactions on beams.
Apply the laws of friction on horizontal and inclined planes.
Use the equations of motion, including vertical motion and projectiles.
Apply Newton's laws to lifts and connected particles.
Use impulse, momentum, work, energy and power.

Key points

Resolve: F cos θ horizontally, F sin θ vertically; resultant √(X² + Y²).
Equilibrium of a particle: ΣFx = 0, ΣFy = 0; Lami for three forces.
Moments: force × perpendicular distance; clockwise = anticlockwise in equilibrium.
Friction: F ≤ μR, F = μR when sliding; on a slope R = mg cos θ.
v = u + at; s = ut + ½at²; v² = u² + 2as; s = ½(u + v)t.
Vertical motion: a = g downward; v = 0 at the top.
Projectile: T = 2u sin θ/g; R = u² sin 2θ/g; H = u² sin²θ/(2g).
F = ma; lift: R = m(g ± a); connected bodies share acceleration.
Impulse = Ft = mv − mu; momentum conserved in collisions.
W = Fs; KE = ½mv²; PE = mgh; P = Fv.

Start here: Forces, resultants and resolving

A force is a vector, so it has magnitude (in newtons, N) and direction. Common forces: weight W = mg acting vertically downward; normal reaction R perpendicular to a surface; tension T along a string; friction F along a surface opposing motion; and applied pushes or pulls.

To find the resultant of several forces, resolve each one into horizontal (x) and vertical (y) components, add each set, then combine.

Force F at angle θ to the horizontal: horizontal component F cos θ, vertical component F sin θ.
Resultant: R = √(X² + Y²), direction tan α = Y/X, where X = ΣFx, Y = ΣFy.

Worked example 19.1

Two forces act at a point: 10 N due east and 8 N at 60° above the east direction. Find the resultant.

X = 10 + 8 cos 60° = 10 + 4 = 14 N.
Y = 0 + 8 sin 60° = 6.928 N.
R = √(14² + 6.928²) = √244 ≈ 15.62 N; α = tan⁻¹(6.928/14) ≈ 26.3° above east.

15.6 N at 26.3° to the 10 N force

Worked example 19.2

Forces of 5 N, 12 N and 13 N act along the directions east, north and south-west respectively. Find the resultant.

South-west is 225° from east. 13 cos 225° = −9.192; 13 sin 225° = −9.192. We use angles measured from east so the signs come out automatically.
X = 5 − 9.192 = −4.192 N; Y = 12 − 9.192 = 2.808 N.
R = √(17.57 + 7.88) ≈ 5.05 N, pointing north-west of west (about 33.8° above the west direction).

≈ 5.05 N

⚠️ Common mistake: adding the magnitudes (5 + 12 + 13 = 30 N). Forces in different directions combine as vectors.

Try it yourself. Find the resultant of 6 N east and 8 N north.

Show solution

10 N at tan⁻¹(8/6) ≈ 53.1° north of east.

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Practice questions

1. Find the resultant of forces 9 N east and 12 N north.

Show worked answer

15 N at 53.1° north of east.

2. A 60 N weight hangs from two strings each at 30° to the horizontal. Find the tension.

Show worked answer

2T sin 30° = 60: T = 60 N.

3. A uniform 5 m plank weighing 150 N rests on its ends; a 250 N load is 1 m from the left end. Find the left reaction.

Show worked answer

Right: (375 + 250)/5 = 125 N; left = 275 N.

4. A 10 kg box on a horizontal floor, μ = 0.4, is pulled by 60 N. Find its acceleration.

Show worked answer

(60 − 40)/10 = 2 m/s².

5. A car decelerates from 30 m/s to rest in 6 s. Find the distance.

Show worked answer

½(30)(6) = 90 m.

Key words

Force: A push or pull, measured in newtons (N).
Resultant: The single force with the same effect as several forces.
Equilibrium: A state with zero resultant force (and zero resultant moment).
Lami's theorem: For three forces in equilibrium, each force is proportional to the sine of the angle between the other two.
Moment: Turning effect: force × perpendicular distance from the pivot.
Couple: Two equal, opposite, parallel forces that cause turning only.
Normal reaction (R): The force a surface exerts perpendicular to itself.
Coefficient of friction (μ): Ratio of limiting friction to normal reaction.
Limiting friction: The maximum friction just before sliding.
Smooth: Frictionless.
Tension: The pulling force in a string.
Projectile: A body moving freely under gravity after being launched.

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